Electron Probability Distribution for the Hydrogen Atom

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The electron in a hydrogen atom is described by the Schrödinger equation. The time-independent Schrödinger equation in spherical polar coordinates can be solved by separation of variables in the form . The radial and angular components are Laguerre and Legendre functions, thus and , respectively. Here, is the first Bohr radius, and are the integers in the ranges (principal quantum number), (angular momentum quantum number), and (magnetic quantum number). The probability of finding an electron at a specific location is given by , where is the normalization constant, such that .


This Demonstration shows the three-dimensional distribution of probability using color when are specified. Only the real part of is considered. Putting the proton at the origin, the probability density is displayed within a sphere of radius . You can select the principal quantum number in the range 1–3, along with the allowed values of the quantum numbers and .


Contributed by: Y. Shibuya (April 2014)
Open content licensed under CC BY-NC-SA



The electron is confined to a space of the order of a few angstroms for . For , the space expands beyond the display limit ( in radius). The patterns for positive and negative values are the same since only the real part of is shown.

Snapshot 1: distribution for ,

Snapshot 2: distribution for ,

Snapshot 3: distribution for ,


[1] J. C. Slater, Modern Physics, New York: McGraw-Hill, 1955.

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